Remember that for the rest of this chapter, it will be assumed that all variables represent non-negative real numbers.
So, for example
= a, a 0.
- A Radical Expression Is Simplified When the Following Are All True
- No perfect powers are factors of the radicand.
- No radicand contains a fraction.
- No denominator contains a radical.
- Statement 1 is accomplished by simplifying radicals as was done in
section 3 of this chapter.
- Statement 2 is accomplished by using the quotient rule and then rationalizing the denominator (see rationalizing the denominator and conjugates, below).
- Statement 3 is accomplished by rationalizing the denominator (see rationalizing the denominator and conjugates, below).
- Quotient Rule For Radicals
- For non-negative real numbers a and b, b 0,
- Rationalizing the Denominator
- When the denominator of a fraction contains a radical, simplify the expression by
- Rationalizing the denominator (discussed next), or
- Using conjugates to rationalize the denominator (discussed below).
- To rationalize a denominator, multiply both the numerator and denominator by a radical that will result in the radicand in the denominator becoming a perfect power.
- Examples of rationalizing the denominator:
- Using Conjugates to Rationalize the Denominator
- When the denominator of an expression is a binomial with at least one of the terms a square root,
then the denominator can be rationalized.
- Do this by multiplying both the numerator and denominator by the conjugate of the denominator.
- The conjugate of a binomial is a binomial having the same two terms with the sign inbetween changed.
- Observe that when conjugates are multiplied together, the square root disappears
- Examples of using conjugates to rationalize the denominator: